Another interesting graph matrix on signed graph is distance matrix.
The definition is given by Hameed, Shijin, Soorya, Germina and Zaslavsky [1]}.
The sign of a path \(P\)  in \(\Sigma \)  is defined as \(\sigma (P)=\Pi _{e\in E(P)}\sigma (e)\) . Let \(P_{(u,v)}\)  denote a shortest path between vertices \(u\)  and \(v\)  and \(\mathcal {P}_{(u,v)}=\lbrace P_{(u,v)}\mid u, v~\mbox{are two given vertices of}~G\rbrace \) . Let \(\sigma _{max}(u,v)=\max \lbrace \sigma (P_{(u,v)})\mid P_{(u,v)}\in \mathcal {P}_{(u,v)}\rbrace \)  and \(\sigma _{min}(u,v)=\min \lbrace \sigma (P_{(u,v)})\mid P_{(u,v)}\in \mathcal {P}_{(u,v)}\rbrace \) . The distance between vertices \(u\)  and \(v\) , i.e., the length of a shortest path between \(u\)  and \(v\) , is denoted by \(d_{uv}\) . Let \(d_{max}(u,v)=\sigma _{max}(u,v)d_{uv}\)  and \(d_{min}(u,v)=\sigma _{min}(u,v)d_{uv}\) . Two types of distance matrices in a signed graph are defined as \(D^{max}(\Sigma )=(d_{max}(u,v))_{n\times n}\)  and \(D^{min}(\Sigma )=(d_{min}(u,v))_{n\times n}\) . Two vertices \(u\)  and \(v\)  in a signed graph \(\Sigma \)  are said to be distance-compatible
(briefly, compatible) if \(d_{max}(u,v)=d_{min}(u,v)\) . And \(\Sigma \)  is said to be (distance-)compatible if every two vertices are compatible. Then
\(D^{max}(\Sigma )=D^{min}(\Sigma )=D^{\pm }(\Sigma )\) .
